Solutions
Name ____________________________
Block _____ Date ________
Algebra: 6.3.1 Association vs. Causation
Bell Work: Graph each parabola.
a. 𝑦 = −(𝑥 + 2)2 + 4
8
b. 𝑦 = (𝑥 + 3)(𝑥 − 3)
y
8
6
6
4
4
2
y
2
x
−8
−6
−4
−2
2
4
6
8
x
−8
−6
−4
−2
2
−2
−2
−4
−4
−6
−6
−8
−8
6-79. Fire hoses come in different diameters. How far the hose can throw
water depends on the diameter of the hose. The Smallville Fire Department
collected data about their fire hoses. The residual plot for the data is shown
at right.
a.
What does the residual plot tell you about the LSRL model
the fire department used?
b.
Find the worst prediction made with the LSRL. How
different was the worst prediction from what was actually
observed? Explain why in context.
c.
Make a conjecture about what the original scatterplot might
have looked like and sketch it. Label both axes.
4
6
8
6-80. The mayor of Smallville finds the following graph in the town’s annual
financial report.
a.
Describe the association, if any, in the scatterplot.
b.
The mayor immediately orders the fire department to send
fewer firefighters to each fire so that there is less damage.
Why do you think the mayor said this? Do you agree with
the mayor’s decision? Explain why or why not.
6-81. A dietician studying the benefits of eating spinach surveyed a large sample of individuals. She recorded
the amount of spinach they ate and their physical strength. The dietician found the spinach eaters to be much
stronger than the non-spinach eaters. The next day the newspaper headline was, “Popeye was right! Eating
spinach makes you stronger!”
a.
Do you agree with the newspaper? Do you agree that if you eat more spinach, you will grow
stronger muscles and increase your strength?
b.
The dietician correctly found an association. What could explain this association other than
spinach makes you stronger?
6-82. A lurking variable is a hidden variable that was not part of the study. The size of the fire in problem 6-80,
and the amount people work out in problem 6-81, are lurking variables.
A medical study found a strong link between the numbers of hours high school students wear a helmet and
the number of concussions (head injuries). However, it is unlikely that wearing helmets causes head
injuries. Can you think of a lurking variable that might explain this association?
6-83. Solve with the quadratic formula: 2𝑥 − 𝑥 2 = −3
𝑥=
−𝑥 2 + 2𝑥 + 3 = 0
−𝑏 ± √𝑏2 − 4𝑎𝑎 −2 ± �22 − 4(−1)(3)
=
2𝑎
2(−1)
𝒙 = 𝟑 and 𝒙 = −𝟏
6-84. Solve by graphing: 3 − 5𝑥 = (𝑥 + 2)2
𝒙 ≈ −𝟖. 𝟗 and 𝒙 = −𝟎. 𝟏
6-85. A human resources manager recorded the experience and hourly wage for a sample of 10 technology
workers.
a.
Create a scatterplot showing the association between the wage and the years of experience in
the calculator. Describe the association.
b.
Is a linear model appropriate? Yes
c.
What is the correlation coefficient? What does it tell you?
R = 0.998.
The linear model is a good fit (R = 1 is perfect)
6-86. Marissa went with her friends to the amusement park on a beautiful spring day. The park was crowded.
Marissa wondered if there was an association between the weather and attendance. From data she received at
the theme park office, Marissa randomly picked ten Saturdays and analyzed the data.
a.
Marissa calculated the LSRL: a = –14 + 0.41t, where a is the attendance
(in 1000s) and t is the high temperature (°F) that day. Interpret the slope
in this context. The population increases approximately 410 people
for every 1 degree increase in temperature
b.
The residual plot Marissa created is shown at right. On days when
temperatures were in the 80s, would you expect the predictions made by
Marissa’s model to be too high, too low, or pretty accurate?
c.
What was the actual attendance on the day when the temperature was 95°F?
6-87. Write an explicit equation for each sequence.
1, 3, 5, 7, … 𝑡(𝑛) = 2𝑛 − 1
a.
1 𝑛
24, 12, 6, 3, … 𝑡(𝑛) = 48 �2�
b.
6-88. Multiply each pair of polynomials.
a.
(2a + b)(a – 3b)
b. (x + 2)(x2 – 2x + 5)
−3𝑏 −6𝑎𝑎 −3𝑏2
𝑎 2𝑎2
𝑎𝑎
2𝑎
+𝑏
+2 2𝑥 2
𝑥
𝑥3
𝑥
2
−4𝑥
10
−2𝑥
+5
−2𝑥 2
5𝑥
6-89. Given the sequence 2, 10, 50, 250, … .
a.
What kind of sequence is it?
b.
Describe the shape of the graph.
c.
Give an explicit equation for the sequence
6-90. Solve each equation. SHOW SUFFICIENT WORK!
a. 3x + 2 = 10 – 4(x – 1)
b. 4(x – 1) – 2(3x + 5) = –3x +1